Patrícia Ewald

(Aspiring) mathematician, gardening enthusiast and plant killer

See my work

About Me

I am a Brazilian master's student in mathematics at the Institute of Mathematics and Statistics of the University of São Paulo (USP), and I earned a bachelor's degree in physics from the Institute of Physics of the same university.

I like to read about and (attempt to) take care of plants, play with my dogs, and play video games. I enjoy learning languages and for the past year I've been taking Japanese lessons (which are now online, unfortunately). In the distant past, I used to study and know some German.

Before the pandemic, I used to enjoy travelling.

My dog, and some books.
Praça do
                                    Relógio, USP, 2020


Here is my cv, and my curriculo Lattes (in Portuguese). During my master's I've been mainly interested in gauge theory in the context of geometric analysis. My thesis is tentatively called 'Compactness in gauge theory' and a short text about it can be found here. My master's advisor is Cristian Ortiz.

In general, I tend to be interested in topics coming from mathematical physics or which exist between different areas of mathematics. Some specific topics I'm interested in (but don't yet know much about, and hope to someday!) are symplectic topology, topological quantum field theory, geometric quantization, and the algebraic geometry side of gauge theory.


Em Agosto de 2021, estou dando um minicurso na III Escola Jayme Tiomno de Física Teórica entitulado "Introdução à Teoria de Gauge Matemática". Vou deixar aqui as notas de aula. Vídeos das aulas estão aqui.

  • Aula 1 - essencialmente álgebra linear. Notas.
  • Aula 2 - vetores tangentes, campos vetoriais, covetores e k-covetores, produto wedge. Notas.
  • Aula 3 - 1-formas e k-formas diferenciais, derivada exterior, comparação com cálculo vetorial, estrela de Hogde no R^3. Notas.
  • Aula 4 - equações de Maxwell em formas diferenciais; intuição e exemplos de fibrados vetoriais. Notas.
  • Aula 5 - mais coment&arios sobre o eletromagnetismo escrito em formas diferenciais (potenciais, transformações de gauge); definições de fibrado vetorial, conexão, curvatura e transformações de gauge, funcional de Yang-Mills e comentários históricos. Notas.


Here are a few texts and presentations I produced.

  • Como a teoria de gauge mudou a física e a matemática (Portuguese/2021) - Slides prepared for a presentation on the impact of gauge theory in physics and mathematics, presented at "Encontro de Egressos e Alunos do Programa de Pós-Graduação em Matemática do IME-USP, 2021".
  • Singular values of compact operators (Portuguese/2020) - Slides prepared for a seminar on singular values of compact operators in Banach spaces, presented for a class on 'Spectral theory on Banach spaces'.
  • A result on Lusternik-Schnirelman theory (2019) - A short text written to clarify the proof of an estimate on the Lusternik-Schnirelman category of a closed subset of a Banach manifold, as given in Palais and Terng's book. Presented as a seminar for a class on 'Calculus of variations and applications to geometry'.
  • The Yang-Mills equation (2019) - A text presenting the physical origins of the Yang-Mills equation, and an introduction to gauge theory. This was written as a final project for a class on 'Fiber bundles and G-structures'.
Below are a couple of study guides I made for myself for classes I took at IME-USP which have turned out to be useful for other people along the years. They are probably only useful for IME-USP students. I mainly made these so I knew what results could be referenced and so I could quickly consult them as I worked on exercises. There may be some mistakes, beware.
  • Cálculo Avançado (2019) - Um compilado de resultados relevantes do livro do Munkres, 'Analysis on Manifolds', que a professora (Vera Carrara) estava seguindo parcialmente quando fiz o curso em 1/2019.
  • Análise Real (2017) - Notas de aula tomadas no curso de Análise Real do professor Rodrigo Bissacot em 2/2017, contendo apenas enunciados e alguns exercícios, sem demonstrações. Está incompleto, eu parei de anotar antes do fim do curso, mas pode ser útil para dar uma ideia rápida do que foi visto no curso.

Cool things

Here are a few things I find cool, useful, or informative for mathematics or life in academia.


I enjoy cool software and programming, and for that reason I like to use some fancy tools with Latex. Here are some:

For more tools (not necessarily Latex related) and some discussion on them, see this blog post by Terence Tao and the comment section.

Not math related

I cannot stress enough how useful Anki can be, especially for vocabulary acquisition when learning a language. It is a memorization tool based on flashcards and SRS. If I were more hardcore I might use supermemo.

Online text editor ilys doesn't let you see what you've written until you've reached a specified word count goal. I find it helpful for overcoming writer's block.